A Stochastic description of eddy-mean flow interactions

When studying large tuburlent regions of the ocean, interactions between the mean flow and eddies plays a
central role in shaping large-scale circulation patterns by redistributing heat, momentum, and energy across the
ocean. Accurately representing these interactions in General Circulation Models (GCMs) remains a challenge,
particularly due to the subgrid-scale modelling issues and the limitations of traditional parameterization methods.
In this study we highlights the limitations of the diagnostics that can be performed with a too small in size en-
semble of simulations for capturing the Reynolds stress tensor as well as accurately diagnosing the work of such
tensor. In the context of studying energy exchange between the mean flow and eddies, the work of the Reynolds
stress is associated with the mean-to-eddy energy conversion rate MEC (Jamet et al. 2022).

To address the above issues, we explore the capabilities of the Location Uncertainty (LU) framework (Mémin
2014) to provide a better representation of eddy-mean flow energy transfer. By introducing stochastic variability
directly into the governing equations of fluid motion, LU provides an approach to model the unresolved turbulent
effects. A rederivation of the equation for the energy transfers is then possible through a stochastic version of
the Reynolds Transport Theorem (Bauer et al. 2020) and leads to an alternative representation of the interactions
between mean flow and eddies.

Based on 48-member ensemble simulation of the North Atlantic under realistic forcing, we provide a robust
comparison between deterministic and stochastic estimates of the work of Reynolds stress (MEC). By comparing
deterministic and stochastic estimates of MEC, we show that LU can effectively address the issues of stastistical
convergence by inflating intrinsic variability leading to a more robust representation of these non-linear terms. In
addition, statistical moments are shown to be more stable than from the deterministic formulation of the eddy-mean
flow interactions. Key results of this study include a detailed formulation of kinetic energy evolution equations
under the LU framework, which reveals significant improvements compared to the deterministic formulation of
the work of the Reynolds stress in terms of statistical moments. The noise definition relies, in this study, on the
snapshot proper orthogonal decomposition (POD) in the ensemble dimension, offering a time varying orthogonal
eigenfunctions basis. These diagnostics provide usefull tools to observe moving patterns and stability regions,
leading to physical interpretation of the eddy-mean flow interactions in the Gulf Stream.